Sure, let's create a test case generator for "Problem I: Train King". This generator will create random valid inputs for the problem following the constraints provided.

Here's the code:

```python
import random

def generate_timetable(num_trains):
    """
    Generate a random timetable for the given number of trains.
    Each train has a random number of cars, random departure, and arrival times.
    """
    timetable = []
    for _ in range(num_trains):
        Ci = random.randint(1, 10)  # Number of cars
        Di = random.randint(0, 86399)  # Departure time
        Ai = random.randint(0, 86399)  # Arrival time
        timetable.append((Ci, Di, Ai))
    return timetable

def tcgen():
    # Generate the number of trains from A to B and from B to A
    NAB = random.randint(0, 50)
    NBA = random.randint(0, 50 - NAB)

    # Generate timetable for trains from A to B
    timetable_AB = generate_timetable(NAB)

    # Generate timetable for trains from B to A
    timetable_BA = generate_timetable(NBA)

    # Prepare the input in the required format
    input_data = f"{NAB} {NBA}\n"
    for train in timetable_AB:
        input_data += f"{train[0]} {train[1]} {train[2]}\n"
    for train in timetable_BA:
        input_data += f"{train[0]} {train[1]} {train[2]}\n"

    return input_data

# Example usage
print(tcgen())
```

This function generates random test cases for the "Train King" problem. It ensures that the number of trains from A to B (`NAB`) and from B to A (`NBA`) are within the specified limits and generates random valid values for the number of cars, departure, and arrival times for each train.

The output is formatted as a string that matches the input format expected by the problem statement. You can use this function to create various test cases to test your solution for the problem.